This title appears in the Scientific Report : 2017 
A diagrammatic derivation of the TAP-approximation
Kühn, Tobias (Corresponding author)
Helias, Moritz
Computational and Systems Neuroscience; INM-6
JARA-BRAIN; JARA-BRAIN
Computational and Systems Neuroscience; IAS-6
2017
Bernstein Conference, Göttingen (Germany), 2017-09-12 - 2017-09-15
Poster
Theory of multi-scale neuronal networks
Supercomputing and Modelling for the Human Brain
Human Brain Project Specific Grant Agreement 1
Theory, modelling and simulation
Connectivity and Activity


Originally invented to describe magnetism, the Ising model has proven to be useful in many other applications, as, for example, inference problems in computer science, socioeconomic physics, the analysis of neural data [1,2,3] and modeling of neural networks (binary neurons). Despite its simplicity, there exists no general solution to the Ising model, i.e. the partition function is unknown in the case of an interacting system. Mean field theory is often used as an approximation being exact in the noninteracting case and for infinite dimensions. A correction term to the mean field approximation of Gibb's free energy (the effective action) of the Ising model was given by Thouless, Anderson and Palmer (TAP) [4] as a “fait accompli” and was later derived by different methods in [5,6,7], where also higher order terms were computed.We present a diagrammatic derivation (Feynman diagrams) of these correction terms and embed the problem in the language of field theory. Furthermore, we show how the iterative construction of the effective action used in the Ising case generalizes to arbitrary non-Gaussian theories.References[1] Tkacik, G., Schneidman, E., Berry II, M. J., Bialek, W. (2008): Ising models for networks of real neurons. arXiv:q-bio/0611072[2] Roudi, Y., Tyrcha, J. and Hertz, J.A. (2009): Ising model for neural data: Model quality and approximate methods for extracting functional connectivity. Phys. Rev. E 79, 051915[3] Hertz, J.A., Roudi, Y. and Tyrcha, J (2011): Ising models for inferring network structure from spike data. arXiv:1106.1752.[4] Thouless, D.J., Anderson, P.W. and Palmer, R.G. (1977): Solution of ’Solvable model of a spin glass’. Phil. Mag. 35 3, 593 – 601[5] Georges, A. and Yedidia, J.S. (1991): How to expand around mean-field theory using high-temperature expansions. J. Phys. A 24, 2173 – 2192[6] Parisi, G. and Potters, M. (1995): Mean-Field equations for spin models with orthogonal interaction matrices. J. Phys. A 28, 5267 – 5285[7] Tanaka, T. (2000): Information Geometry of Mean-Field Approximation. Neur. Comp. 12, 1951-1968.Acknowledgements. This work was partially supported by HGF young investigator’s group VH-NG-1028, Helmholtz portfolio theme SMHB, Juelich Aachen Research Alliance (JARA), and EU Grant 604102 (Human Brain Project, HBP).